A stage extraction process is depicted in Fig. P$12\mathrm{.}9.$ Ih such systems, a stream containing a weight fraction $Y_{in}$ of a chemical enters from the left at a mass flow rate of $F_1.$ Simultaneously, a solvent carrying a weight fraction $x_{in}$ of the same chemical enters from the right at a flow rate of $F_2.$ Thus, for stage $i,$ a mass balance can be represented as

$F_1{Y}_{i-1}+F_2{x}_{i+1}=F_1{Y}_i+F_2{x}_i$

At each stage, an equilibrium is assumed to be established between $Y_i$ and $x_i$ as in

$K=\frac{x_i}{Y_i}$

where $K$ is called a distribution coefficient. Equation $($P$12\mathrm{.}9\mathrm{.}2)$ can be solved for $x_i$ and the result substituted into Eq$.($P$12\mathrm{.}9\mathrm{.}1)$ to yield

$Y_{i-1}-(1+\frac{F_2}{F_1}K)Y_i+(\frac{F_2}{F_1}K)Y_{i+1}=0$

If $F_1=400k\frac{g}{h},Y_{in}=0\mathrm{.}1,F_2=800k\frac{g}{h},x_{in}=0,$ and $K=0\mathrm{.}2,$ determine the values of $Y_{\text{o}\text{u}\text{t}}$ and $x_{\text{o}\text{u}\text{t}}$ using $1$ to $15$ stages and plot them vs the number of stages. Note that Eq$.($P$12\mathrm{.}9\mathrm{.}3)$ must be modified to account for the inflow weight fractions when applied to the first and last stages.